Edith Keeler's death in The City on The Edge of Forever, the interactions of Kirk, Spock and McCoy in that scene are the best of the entire series.

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Whatever the behind-the-scenes disputes doing on with that episode, the actors did a wonderful job.

And my #1 tear-jerker moment is, of course, Spock's death and funeral. I don't get why so many people in the audience laughed when Scotty played Amazing Grace. I'm atheist, and that song always reduces me to tears.

The destruction of the Enterprise in ST III. (I said "My God" at the exact same moment as Kirk there. My wife whispered, "How did you know he was going to say that?" "I didn't." I told her with tears in my eyes. I actually didn't think they would do it).

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It was (unfortunately) in the trailer. How did you manage to avoid that before seeing the film?

The destruction of the Enterprise in ST III. (I said "My God" at the exact same moment as Kirk there. My wife whispered, "How did you know he was going to say that?" "I didn't." I told her with tears in my eyes. I actually didn't think they would do it).

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It was (unfortunately) in the trailer. How did you manage to avoid that before seeing the film?

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Back in the 80's I was manager of a shipping department of a carpet mill. I worked 80 to 90 hours a week, had no internet, watched little television besides the news so I must have missed any tv ads showing this. I only knew the release day due to the news paper listing. So I was able to see all the TOS movies on their opening day with zero spoilers of any kind, other than the titles themselves.

The only thing in Trek, and I've seen every episode and movie and least once, that's ever brought a tear to my eye is Spock's death in TWOK. It was just done perfectly, two friends, the love, the loyalty, separated, unable to even touch... Just perfectly done.

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I agree, it was perfect. But there are others that bring tears to my eyes.

TNG:

the end of The Inner Light, and Picard and Daren's duet in Lessons
Lal's death in The Offspring
Tasha Yar's funeral in Skin of Evil
Geordi and Ro Laren's funeral (or was it just being planned? I don't remember) in The Next Phase, and RL at the end of Rascals

DS9:

Benny's breakdown in Far Beyond the Stars
Ziyal's death in Sacrifice of Angels
(There were probably more, but I haven't watched DS9 in a very long time.)

Other top contenders (I can cry easily):
TOS: "Let me help" speech.
Spock's death
Lal's death
"It's me" speech from Inner Light
Data's death
"It's not linear" speech from the Emmisary
"I'm a good tailor" speech from The Die is Cast
End of The Visitor
"I need a miracle. Bajor needs a miracle"
"Maybe a year. Maybe yesterday."

1. Lal & Data
2. Jadzia...
3. Spock dying
4. Spock's funeral
5. The Inner Light
6. The breakdown in, 'Far Beyond the Stars'.
7. Neelix leaving at the end of Voyager.
8. Tasha's funeral.
9. T'Pol and Trip dealing with the death of Elizabeth.
10. B'elanna and Tom revealing their feelings for each other while drifting in space.
11. B'elanna revealing why she wanted to change the looks of her child.